New York J. Math. 1(1995)196–205.
Cohomology of Modules in the Principal Block of a Finite Group
D. J. Benson
Abstract. In this paper, we prove the conjectures made in a joint paper of the author with Carlson and Robinson, on the vanishing of cohomology of a finite groupG. In particular, we prove that ifkis a field of characteristicp, then every non-projectivekG-moduleM in the principal block has nontrivial cohomology in the sense thatH∗(G, M)6= 0, if and only if the centralizer in Gof every element of orderpisp-nilpotent (this was proved forpodd in the above mentioned paper, but the proof here is independent ofp). We prove the stronger statement that whether or not these conditions hold, the union of the varieties of the modules in the principal block having no cohomology coincides with the union of the varieties of the elementary abelianp-subgroups whose centralizers are notp-nilpotent (i.e., the nucleus). The proofs involve the new idempotent functor machinery of Rickard.
Contents
1. Introduction 196
2. Terminology and Background Material 199
3. Inducing Idempotents 200
4. An Equivalence of Categories 201
5. The Main Theorems 202
References 205
1. Introduction
Recent developments in modular representation theory of finite groups have in- volved a re-evaluation of the role of infinitely generated modules. In particular, Rickard [5] has introduced some infinitely generated modules which are idempotent in the stable category, in the sense that the tensor square is isomorphic to the orig- inal module plus a projective. This work, together with a version of Dade’s lemma for infinitely generated modules, has allowed Benson, Carlson and Rickard [1,2] to
Received March 15, 1995.
Mathematics Subject Classification. 20C20, 20J06.
Key words and phrases. finite group, representations, cohomology, nucleus, idempotent functor.
Partially supported by a grant from the NSF.
1995 State University of New Yorkc ISSN 1076-9803/95
196
formulate and prove a generalization of the usual theory of complexity and varieties for modules to the infinitely generated situation.
In this paper, we shall demonstrate how the recent work described above can be used to address some older questions about finitely generated modules. In particu- lar, we shall prove the conjectures formulated in the paper of Benson, Carlson and Robinson [3]. Before stating our main theorem, we state a more easily understood consequence, which provides an affirmative solution to Conjecture 1.2 of that paper.
The proof may be found in Section5.
Theorem 1.1. Let k be a field of characteristic p, and let G be a finite group.
Then the centralizer of every element of orderp inGis p-nilpotent, if and only if for every non-projective module M in the principal block B0(kG),Hn(G, M)6= 0 for somen6= 0.
We remark that in general, for akG-moduleM,Hn(G, M)6= 0 for somen6= 0 if and only if ˆHn(G, M) 6= 0 for infinitely many values of n both positive and negative, cf. Theorem 1.1 of [3]. We also remark that in the statement of the above theorem, it does not matter whether we restrict our attention to finitely generated kG-modules.
In the language of [3], our main theorem is the following. This almost provides an affirmative answer to Conjecture 10.10 of that paper, which does not mention passing down to summands. Terminology used in this introduction is explained in Section2, and the proof may be found in Section5.
Theorem 1.2. Let k be an algebraically closed field of characteristicp, and letG be a finite group. Then every finitely generatedkG-module in the principal block is a direct summand of a nuclear homology module.
It follows from this theorem that every kG-module in the principal block is a filtered colimit of nuclear homology modules. In order to prove that every finitely generated module in the principal block actually is a nuclear homology module, it would suffice to show that the characters of nuclear homology modules span the principal block, and then the argument would run as in the proof of Proposition 4.4 of [3]. It does not seem immediately clear that this character theoretic statement is true.
As pointed out in Corollary 10.12 of [3] (passage to direct summands does not affect this), it follows from this theorem that the nucleus YG (which is the union of the images in the cohomology varietyVG, of the elementary abelianp-subgroups whose centralizers are not p-nilpotent) coincides with the representation theoretic nucleus ΘG (which is the union of the varieties of the finitely generated modules in the principal block having no cohomology).
Corollary 1.3. For any finite groupG, we haveYG= ΘG.
In the case whereYG={0}, this implies Theorem1.1. More precisely, we prove the following strengthened form of Theorem 1.4 of [3]:
Theorem 1.4. Suppose that G is a finite group and k is a field of characteristic p. Then the following are equivalent:
(A) Every finitely generated module in the principal blockB0(kG)is a trivial ho- mology module.
(A0) Every simple module in B0(kG) is a direct summand of a trivial homology module.
(A00) Every(finitely generated)trivial source module inB0(kG)is a direct summand of a trivial homology module.
(A000) Every module(finitely or infinitely generated) inB0(kG) is a filtered colimit of trivial homology modules.
(B) For every finitely generated non-projective module M in B0(kG), we have Hn(G, M)6= 0 for somen >0.
(B0) For every finitely generated non-projective periodic moduleM inB0(kG), we haveHn(G, M)6= 0 for somen >0.
(B00) For every(finitely or infinitely generated)non-projective moduleM inB0(kG), we have Hn(G, M)6= 0for somen >0.
(B000) For every (finitely or infinitely generated) module M of complexity one in B0(kG), we haveHn(G, M)6= 0 for somen >0.
(C) For every non-projective(finitely generated)trivial source moduleM inB0(kG), we have Hn(G, M)6= 0for somen >0.
(D) The centralizer of every element of order pin Gisp-nilpotent.
(D0) The centralizer of every nontrivial p-subgroup of Gisp-nilpotent.
(E) For every non-projective finitely generated indecomposable moduleM inB0(kG), with vertexRand Green correspondentf(M), we haveHn(NG(R), f(M))6= 0 for somen >0.
On the way to proving these theorems, we prove a remarkable property of mod- ules of complexity one. In general, such a module decomposes as a direct sum of modules whose variety consists of a single line through the origin in the cohomology varietyVG(k). The following theorem is proved at the end of Section3.
Theorem 1.5. Suppose thatM is akG-module whose varietyVG(M)consists of a single lineLthrough the origin inVG(k). LetEbe an elementary abelianp-subgroup of G, minimal with respect to the property thatL is contained in the image of the map res∗G,E : VE(k)→ VG(k) induced by restriction from G to E in cohomology.
Let L = res∗G,E(`0) with `0 a line through the origin in VE(k), and let D be the subgroup ofNG(E)consisting of the elements which stabilize`0 setwise. Then the direct sum ofM with some projectivekG-module is induced from D.
We remark that this paper makes Sections 8 and 9 of [3] obsolete, and there is no longer anything special about odd primes in our proofs. We also remark that we have not been able to tackle the original question which motivated [3], namely whether every simple module in the principal block necessarily has nonvanishing cohomology. One possible approach to this might be to try to prove that the variety of a simple module in the principal block cannot be contained in the nucleus. Since the property of being simple is not easy to work with, it may be better to consider modules whose endomorphism rings, modulo traces from suitable subgroups, are isomorphic to the field.
I would like to thank Jon Carlson and Jeremy Rickard for conversations which inspired this work, and I would also like to thank Jon Carlson for pointing out a serious error in an earlier version of this paper.
2. Terminology and Background Material
LetGbe a finite group andkan algebraically closed field of characteristicp. Let stmod(kG) be the stable category of finitely generated kG-modules, considered as a triangulated category. The homomorphisms HomkG(M, N) in this category are homomorphisms in the usual module category, modulo those that factor through some projective module. The triangles instmod(kG) come from the short exact se- quences inmod(kG) in the normal way. Similarly,StMod(kG) is the stable category of all (not necessarily finitely generated)kG-modules, which is again a triangulated category.
We write VG(k) for the maximal ideal spectrum of H∗(G, k). Note that for p odd, elements of odd degree square to zero, so thatH∗(G, k) modulo its nil radical is commutative. Thus VG(k) is a homogeneous affine variety. Associated to any (not necessarily finitely generated) kG-moduleM, there is a collection VG(M) of closed homogeneous irreducible subvarieties of VG(k) (see [2] for details). These varieties have good properties with respect to tensor products, andM is projective if and only ifVG(M) =∅.
We next remark that there is a mistake in the definition ofYGgiven in Section 10 of [3]. Thenucleus YG should be defined as the subvariety ofVG(k) given as the union of the images of the maps res∗G,H : VH(k) → VG(k) induced by resG,H : H∗(G, k)→H∗(H, k), asHruns over the set of subgroups ofGfor whichCG(H) is notp-nilpotent (and not the union of the images of res∗G,CG(H):VCG(H)(k)→VG(k) as stated there; also in the proof of Theorem 10.2 of that paper,VCG(H)should be replaced byVH, and no other changes are necessary).
Therepresentation theoretic nucleusΘG is the subset ofVG(k) given as the union of the varietiesVG(M) asM runs over the finitely generated modules in the principal block B0(kG) with Hn(G, M) = 0 for all n. By Theorem 6.4 of [3], it suffices to consider periodic modules in this definition.
We say that a finitely generatedkG-moduleM is atrivial homology module or a TH module if there exists a finite complex (Ci, δi : Ci → Ci−1) of finitely generatedkG-modules and homomorphisms such that the following conditions hold:
(i) EachCi is a projectivekG-module, andCi= 0 fori <0 and forisufficiently large.
(ii) Fori >0,Hi(C∗) is a direct sum of copies of the trivialkG-modulek.
(iii) H0(C∗)∼=M.
We say that a finitely generatedkG-moduleM is anuclear homology module or anNH moduleif it satisfies the same conditions, but with (ii) replaced by:
(ii0) For i > 0, Hi(C∗) is a direct sum of copies of the trivial kG-module k and finitely generated modulesM0 in B0(kG) withVG(M0)⊆YG.
We write T H and N H for the thick subcategories ofstmod(kG) consisting of the direct summands of trivial homology modules and of nuclear homology modules respectively.
Next, we recall from Section 5 of Rickard [5] that given any thick subcategoryC ofstmod(kG), there are functorsEC andFC onStMod(kG) satisfying the following properties:
(a) For anyX inStMod(kG),EC(X) is a filtered colimit of objects in C.
(b) For anyX inStMod(kG),FC(X) isC-local, in the sense that for any object M in C, HomkG(M,FC(X)) = 0.
(c) There is a triangle inStMod(kG)
EC(X)→X → FC(X)→Ω−1EC(X).
In fact (see the remark after Proposition 5.7 of [5]) the functors EC andFC are characterized by these properties.
Our goal will be to show that the functorFN His the zero functor on the principal block, which will enable us to prove that every finitely generated module in the principal block is a direct summand of a nuclear homology module.
IfV is a closed homogeneous subvariety ofVG(k), we writeCV for the subcategory of stmod(kG) consisting of the finitely generated modules M with VG(M) ⊆ V. Then the corresponding functorsEV =ECV andFV =FCV are given by tensoring with certain (usually infinitely generated) moduleseV andfV. These are orthogonal idempotents inStMod(kG), in the sense thateV⊗eV ∼=eV⊕(projective),fV⊗fV ∼= fV ⊕(projective), andeV ⊗fV is projective. The triangle for a moduleX in this situation is given by tensoring the triangle
eV −→λV k−→µV fV →Ω−1eV
withX.
3. Inducing Idempotents
LetLbe a line through the origin in VG(k). Then by the Quillen stratification theorem, there is an elementary abelianp-subgroupE, uniquely determined up to conjugacy, with the property that L is in the image of res∗G,E : VE(k) → VG(k), but Lis not in the image of res∗G,E0 :VE0(k)→VG(k) for any proper subgroupE0 ofE. In this situation, we say that Loriginates inE. We write C=CG(E) for the centralizer, and N = NG(E) for the normalizer in G of E. Let `0 be a line through the origin in VE(k) with L = res∗G,E(`0), and let D be the subgroup of NG(E) consisting of the elements which stabilize `0 setwise. Then `0 and D are uniquely determined up to conjugacy inN. Since`0originates inE, the centralizer C is equal to the pointwise stabilizer of `0. Any finite group of automorphisms of the line `0 is cyclic of order prime to p, so we have C ED ≤N with D/C a cyclic group of order prime to p. Finally, we set` = res∗D,E(`0)⊆VD(k), so that L= res∗G,D(`).
Theorem 3.1. With the above notation, let e` be the idempotentkD-module cor- responding to` andeL be the idempotent kG-module corresponding toL. Then
e`↑G∼=eL⊕(projective).
Proof. Consider the composite map
e`↑G−−−−−→λ`↑G kD↑G ν−→k,
where ν :kD↑G→k is the augmentation map. On restriction toE, this becomes (modulo projectives) the composite map
M
g∈N/D
g⊗e`↓E→
M
g∈N/D
g⊗k
⊕(induced modules)−→ν↓E k.
Here, g ∈ N/D means that g runs over a set of left coset representatives of D in N. The first map is the sum of all the maps
g⊗e`↓E ∼= eg(`)↓E
↓ ↓
g⊗k ∼= k and the second map ν ↓E sends P
igi⊗λi to P
iλi. On restriction to a cyclic shifted subgroup corresponding to a point in`0, the summandsg⊗e`↓E forg6∈D give projective modules, while 1⊗e` restricts to give k⊕(projective), because`0
isn’t fixed by anyg∈N\D. Moreover, this copy ofkmaps isomorphically to 1⊗k in the second module, and then isomorphically tok in the third module. So if we complete to a triangle
e`↑G→k→f →Ω−1e`↑G, thenf restricted to this cyclic shifted subgroup is projective.
The module e`↑G is a filtered colimit of modules in CL, since e` is a filtered colimit of modules inC`. ForM inCL, Homk(M, f) is projective, by a combination of Dade’s lemma (the infinite dimensional version given in Section 3 of [2]) and Chouinard’s theorem [4], sof isCL-local.
By Rickard’s characterization (see the remark after Proposition 5.7 of [5]), the triangle
e`↑G→k→f →Ω−1e`↑G is isomorphic to
eL →k→fL →Ω−1eL. Corollary 3.2. If M is a module whose variety VG(M) = {L} with L as above, thenM ⊕(projective)is induced fromD.
Proof. IfVG(M)⊆ {L} then using the theorem, we have
M⊕(projective)∼=M ⊗eL∼=M ⊗e`↑G∼= (M↓D⊗e`)↑G,
and soM ⊕(projective) is induced fromD.
This completes the proof of Theorem1.5.
4. An Equivalence of Categories
We can combine the results of the last section with the Mackey decomposition theorem to obtain an equivalence of categories as follows. LetCbe the full subcate- gory ofStMod(kG) consisting ofkG-modulesM withVG(M)⊆ {L}(or equivalently M ∼=eL⊗M), and letC0 be the full subcategory ofStMod(kD) consisting of mod- ules M0 with VD(M0) ⊆ {`} (or equivalently M0 ∼=e`⊗M0). Using the Mackey decomposition theorem, we see that ifM0 is inC0 thenM0↑G↓D is isomorphic to a direct sum ofM0 with a module M00satisfyingVD(M00)∩ {`}=∅. So we have
e`⊗(M0↑G↓D)∼=M0.
Since every object inCis induced from an object inC0 by Corollary 3.2, it follows that the functors (e`⊗ −)◦resG,D : C → C0 and indD,G : C0 → C are mutually inverse equivalences of categories.
Lemma 4.1. If M is akG-module inC, which lies in the principal blockB0(kG), thene`⊗M↓Dis a direct sum of a projective module and a module in the principal block B0(kD).
Conversely, ifM is a kG-module inC with no summand in the principal block B0(kG), thene`⊗M↓D is a direct sum of a projective module and a module with no summand in the principal blockB0(kD).
Proof. Let e be a block idempotent of kG, and let BrE : Z(kG) → Z(kD) be the Brauer map with respect toE. Ifb is any block ofkD, say with defect group R, then E ≤ R ≤ C, and so RCG(R) ≤ C. So the Brauer correspondent bG is defined, and by Brauer’s third main theorem, bG is equal toB0(kG) if and only if b=B0(kD). It follows that ife0is the principal block idempotent ofkGande1 is the principal block idempotent ofkD, then BrE(e0) =e1, and BrE(e)e16= 0 if and only ife=e0.
IfM is a finitely generatedkG-module withe.M=M, then Nagao’s lemma says that
M↓D∼= BrE(e).M↓D⊕M1
where M1 is a direct sum of modules which are projective relative to subgroups Q≤CwithE6≤Q. Since the variety ofe`⊗M↓Dhas trivial intersection with the image ofVE0 →VD for any proper subgroupE0 ofE, it follows that
e`⊗M↓D∼= BrE(e).(e`⊗M↓D)⊕M2 whereM2is projective.
If M =e.M is not finitely generated, express it as a filtered colimit of finitely generated modules Mα in C. Eache`⊗Mα↓D may be written as a direct sum of BrE(e).(e`⊗Mα↓D) and a projective module killed by BrE(e). There are no maps between these two types of summands, so when we pass to the colimit, we obtain
a decomposition ofe`⊗M↓D of the desired form.
Theorem 4.2. The functors(e`⊗−)◦resG,D:C → C0 andindD,G:C0→ Care mu- tually inverse equivalences of categories, and induce mutually inverse equivalences between the full subcategories B0(kG)∩ C andB0(kD)∩ C0.
Proof. This follows immediately from the lemma and the discussion preceding
it.
5. The Main Theorems
We continue with the same notation. Namely,L is a line through the origin in VG(k) originating in an elementary abelianp-subgroupEofG. We setC=CG(E), N =NG(E) andL= res∗G,E(`0), with `0 a line through the origin in VE(k). We set D equal to the stabilizer in N of the line `0. We set `1 = res∗C,E(`0)⊆VC(k) and`= res∗D,E(`0)⊆VD(k).
Lemma 5.1. Suppose that C isp-nilpotent. Then for any module M0 inB0(kD) satisfyingVD(M0) ={`}, we have Hˆn(D, M0)6= 0 for somen.
Proof. The argument for this is given in the proof of Proposition 6.8 of [3]; we repeat it here for convenience. Let ¯C = C/Op0(C) and ¯D = D/Op0(C). Then C¯ is a p-group, and ¯D/C¯ is a cyclic p0-group. By Lemma 6.7 of [3], we may choose a homogeneous element ζ ∈ Hm( ¯C, k) = Hm(C, k) for some m, so that
`1 ∩VC(hζi) = {0}, and so that the one dimensional subspace hζi ⊆ H∗(C, k) is ¯D/C-invariant and affords a faithful one dimensional representation of ¯¯ D/C.¯ For a suitable one dimensional representation ε of D with kernel C, ζ may be regarded as an element of ExtmkD(k, ε). Thusζ is represented by a homomorphism ζˆ : Ωm(k) → ε, and we write Lζ for the kernel of ˆζ. So there is a short exact sequence ofkD-modules
0→Lζ →Ωm(k)→ε→0.
Tensoring withM0, we obtain a short exact sequence
0→Lζ⊗M0 →Ωm(k)⊗M0→ε⊗M0→0.
The tensor product theorem for varieties (Theorem 10.8 of [2]) implies thatLζ⊗M0 is projective, and so we obtain a stable isomorphism Ωm(M0)∼=ε⊗M0.
SinceM0 is non-projective, for some value ofrwe have Extd0kD(εr, M0) = HomkG(εr, M0)6= 0,
where εr denotes the rth tensor power of ε. This is because every simple module inB0(kD) is isomorphic to some suchεr. Thus
Hˆmr(D, M0)∼= ˆH0(D,Ω−mr(M0))∼= ˆH0(D, ε−r⊗M0)∼=Extd0kD(εr, M0)6= 0.
Here,ε−rdenotes therth tensor power of the dual module ε∗. Theorem 5.2. Suppose that M is a module in B0(kG) with VG(M) ={L}, and that C isp-nilpotent. ThenHˆn(G, M)6= 0for somen.
Proof. By Theorem 4.2, there is a module M0 in B0(kD) with M0 ↑G∼= M ⊕ (projective). By Shapiro’s lemma we have ˆHn(G, M)∼= ˆHn(D, M0). By Lemma5.1,
this is nonzero for somen.
Corollary 5.3. Suppose that M is a non-projective kG-module in B0(kG) with the property thatVG(M)contains no closed homogeneous subset of the nucleus YG. ThenHˆn(G, M)6= 0 for somen.
Proof. We use the argument given in Theorem 6.4 of [3] to reduce to the complex- ity one case. LetKbe an algebraically closed extension ofkof large transcendence degree. Since M is non-projective, VG(M) contains a closed homogeneous irre- ducible subsetV which is not contained inYG. So VG(K⊗kM) contains a generic lineLforV. Choose elementsζ1, . . . , ζs∈H∗(G, K) so that
VG(K⊗kM)∩ VGhζ1i ∩ · · · ∩ VGhζsi={L}.
Here, VGhζii is the collection of closed homogeneous subsets of the hypersurface VGhζiidefined byζi. Then we have
VG((K⊗kM)⊗KLζ1⊗K· · · ⊗KLζs) ={L}.
Next, we note that in Lemma 6.3 of [3], although M2 needs to be finitely gen- erated,M1 does not. So every non-projective summand ofM1⊗Lζ is in the same
block asM1. So every non-projective summand of (K⊗kM)⊗KLζ1⊗K· · · ⊗KLζs
is inB0(KG), and by the theorem, we have
Hˆn(G,(K⊗kM)⊗KLζ1⊗K· · · ⊗KLζs)6= 0 for infinitely many values ofn, positive and negative.
Similarly, in Lemma 6.2 of [3], althoughM1must be finitely generated,M2need not be. So ifζis a homogeneous element in cohomology, then ˆHn(G, M2) = 0 for all nimplies ˆHn(G, Lζ⊗M2) = 0 for alln. So we may deduce that ˆHn(G, K⊗kM)6= 0 for infinitely many values ofn, positive and negative. Finally, this implies that the
same is true of ˆHn(G, M).
Proposition 5.4. IfM is anN H-localkG-module, thenVG(M)contains no closed homogeneous subset of the nucleusYG.
Proof. IfVG(M) contains a closed homogeneous subsetV ofYG, then HomkG(EV(M), M)6= 0,
while ifM isN H-local,EN H(M) = 0. However, any map fromEV(M) toM factors throughEN H(M), because the subcategory ofstmod(kG) consisting of modules with
variety inV is contained inN H.
Theorem 5.5. If M is a module inB0(kG), thenEN H(M)∼=M andFN H(M) = 0.
Proof. Consider the variety ofFN H(M). By Proposition5.4, it contains no closed homogeneous subset of the nucleus YG. So if FN H(M) is nonzero in StMod(kG) (i.e., non-projective), its variety must contain some closed homogeneous subset which is not in the nucleus. Then by Corollary5.3, we have ˆHn(G,FN H(M))6= 0 for infinitely many values ofn. So for somen, we have HomkG(Ωn(k),FN H(M))6=
0. Since Ωn(k) is an NH module, this contradicts the fact that FN H(M) is N H- local. It follows thatFN H(M) = 0, and therefore thatEN H(M)∼=M. Proof of Theorem 1.2. By Theorem5.5, ifM is inB0(kG), thenEN H(M)∼=M. SoM is a filtered colimit of NH modules, and since it is finitely generated, it follows
that it is a direct summand of an NH module.
Proof of Corollary 1.3. It is shown in Corollary 10.12 of [3] that this follows
from Theorem1.2.
Theorem 5.6. Suppose that the centralizer of every element of order p in G is p-nilpotent. Then every finitely generated module in the principal block is a trivial homology module.
Proof. The condition on G is equivalent to the condition that YG = {0}. So under these conditions, nuclear homology modules are the same as trivial homol- ogy modules. So the theorem follows from Theorem 1.2, using Theorem 3.5 and
Propositions 4.4 and 4.5 of [3].
Proof of Theorem 1.4. It is proved in [3] that (A) ⇔ (A0) ⇔ (A00) ⇒ (B) ⇔ (B0)⇒(C) ⇔(D) ⇔(D0)⇔(E). It is clear that (A000) ⇒(A0), (B00)⇒(B) and (B00)⇒(B000)⇒(B0). Theorem5.6shows that (D)⇒(A000). Finally, Corollary5.3
shows that (D)⇒(B00).
Proof of Theorem 1.1. This is just the statement that (B00) ⇔ (D) in Theo-
rem1.4, so this is now proved.
References
[1] D. J. Benson, J. F. Carlson and J. Rickard.Complexity and varieties for infinitely generated modules,To appear in Math. Proc. Camb. Phil. Soc.
[2] D. J. Benson, J. F. Carlson and J. Rickard.Complexity and varieties for infinitely generated modules,II, Preprint, 1995.
[3] D. J. Benson, J. F. Carlson and G. R. Robinson. On the vanishing of group cohomology, J. Algebra 131 (1990), 40–73.
[4] L. Chouinard.Projectivity and relative projectivity over group rings,J. Pure Appl. Algebra 7 (1976), 278–302.
[5] J. Rickard.Idempotent modules in the stable category,Preprint, 1994.
Department of Mathematics, University of Georgia, Athens GA 30602, USA.
